Polar perturbations of dilaton-Euler-Heisenberg black holes

We investigate the quasinormal modes of polar metric-dilaton perturbations around the dilaton-Euler-Heisenberg (dEH) black holes with dilaton hair. The dEH black holes are obtained from the Einstein-Maxwell-dilaton theory with two dilaton coupling parameters ($\alpha,\beta$) to the nonlinear Euler-Heisenberg term. We compute the quasinormal mode spectra by making use of two numerical techniques: direct integration and matrix values continued fraction methods. An excellent agreement is found between two approaches, confirming the robustness of our computation. We present the fundamental quasinormal frequencies for both gravitational and dilaton modes and analyze their dependence on the magnetic charge ($Q_m$), angular momentum quantum number ($l$), and coupling parameter ($\epsilon=\alpha-\beta$). All negative imaginary quasinormal frequencies for polar metric-dilaton perturbations imply that the dEH black hole with dilaton hair is stable against dilaton with $l=0,1,2,3$ and gravitational modes with $l=2,3$. Also, our results reveal distinct qualitative behaviors between $\epsilon=1$ and $\epsilon=-1$, particularly in the damping rates near the extremality.